Theorem seqeq1 10595

Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Assertion

Ref	Expression
seqeq1	|- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Proof of Theorem seqeq1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 |- ( M = N -> M = N )
2		fveq2 5576	. . . . . 6 |- ( M = N -> ( F ` M ) = ( F ` N ) )
3	1, 2	opeq12d 3827	. . . . 5 |- ( M = N -> <. M , ( F ` M ) >. = <. N , ( F ` N ) >. )
4		freceq2 6479	. . . . 5 |- ( <. M , ( F ` M ) >. = <. N , ( F ` N ) >. -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
5	3, 4	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
6		fveq2 5576	. . . . . 6 |- ( M = N -> ( ZZ>= ` M ) = ( ZZ>= ` N ) )
7		eqid 2205	. . . . . 6 |- _V = _V
8		mpoeq12 6005	. . . . . 6 |- ( ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ _V = _V ) -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
9	6, 7, 8	sylancl 413	. . . . 5 |- ( M = N -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
10		freceq1 6478	. . . . 5 |- ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
11	9, 10	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
12	5, 11	eqtrd 2238	. . 3 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
13	12	rneqd 4907	. 2 |- ( M = N -> ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
14		df-seqfrec 10593	. 2 |- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
15		df-seqfrec 10593	. 2 |- seq N ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. )
16	13, 14, 15	3eqtr4g 2263	1 |- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2772   <.cop 3636   ran crn 4676   ` cfv 5271  (class class class)co 5944   e. cmpo 5946  freccfrec 6476   1c1 7926   + caddc 7928   ZZ>=cuz 9648   seqcseq 10592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fv 5279  df-oprab 5948  df-mpo 5949  df-recs 6391  df-frec 6477  df-seqfrec 10593

This theorem is referenced by:  seqeq1d  10598  seq3f1olemqsum  10658  seqf1oglem2  10665  seq3id  10670  seq3z  10673  iserex  11650  summodclem2  11693  summodc  11694  zsumdc  11695  isumsplit  11802  ntrivcvgap  11859  ntrivcvgap0  11860  prodmodclem2  11888  prodmodc  11889  zproddc  11890  fprodntrivap  11895  ege2le3  11982  gsumfzval  13223  gsumval2  13229